Assignment 2: M5210 Stochastic Calculus and Mathematical Finance
Due in June 12 on Moodle
Bt is a Brownian motion, B0 = 0.
1. Let the price process Xt be a diffusion with no drift coefficient.
Let the claim at time T be CT = g(XT), and its price at time t
be Ct = E(g(XT)|Ft). Find the hedging of this claim, ie. find
a predictable process Ht such that Ct = t HtdXt. Find explicit 0
expression for Ht when the diffusion coefficient σ(x) = σx, and g(x) = (x − K)+.
2. Derive the PDE for the price of the option in Heston model.
Hint: A way to derive a PDE for the option price is based on the fact that C(t)e−rt is a Q-local martingale. Denote C(S, v, t) the option price, expand d(C(S(t), v(t), t)e−rt) and equate the coefficient of dt to zero.
Heston’s model under Q
dS(t) = rS(t)dt + v(t)S(t)dB(t)
dv(t) = α(μ − v(t))dt + δv(t)dW (t),
where B and W are correlated Brownian motions, correlation ρ.
3. St is a stock price which is a positive Itˆo process, S0 = 1, and βt = ert represent a savings account, 0 ≤ t ≤ 1. Let a(t) = 1 (1−e−r(1−t))
r
and b(t) = e−r t Sudu. Show that Vt = a(t)St + b(t)βt is a self-
0
financing portfolio. Give the value of this portfolio at time T = 1.
4. Let St = 1 + Bt and βt = et (Bachelier model). Give the change of measure Q such that St/βt is a Q martingale. Give SDE for St under Q.
Hint: Use that E( BtdBt) is a martingale.
5. In the Vasicek model the spot rate satisfies stochastic differential equation
drt =b(a−rt)dt+σdBt, 0≤t≤T, r0 >0,
where Bt is a P-Brownian motion. Let Λ = eλBT −λ2T/2, for some λ > 0, and the measure Q defined by dQ/dP = Λ. Give the equation for rt under Q.
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